In this section, we consider the “base case” of mode estimation, in which \(\mathcal {P}\)  takes exactly \(K = 2\)  values. Notice that \(\mathcal {P}(p, v, 2)\)  is a Bernoulli distribution that generates \(v_{1}\)  with probability \(p_{1}\)  and \(v_{2}\)  with probability \(p_{2} = 1 - p_{1}\) . Treating \(p_{1} \in [0, 1]\)  as the sole parameter of the distribution, our task is to devise a \(\delta \) -correct stopping rule to test if \(p_{1} > \frac{1}{2}\) . Since \(p_{1}\)  may be arbitrarily close to \(\frac{1}{2}\) , it is not possible to decide beforehand how many samples suffice for the test to succeed. An unfortunate consequence of having a random stopping time is that it cannot be used directly within concentration inequalities such as Chernoff bounds. Rather, stopping rules invariably go through a union bound over all possible stopping times, dividing the mistake probability \(\delta \)  among them [1]}, [2]}. Although there has been progress towards optimising this apportioning of \(\delta \)  [3]}, [4]}, resulting methods still have tunable parameters in their “decay rates” that govern the stopping time.
